UNIVERSAL INEQUALITIES FOR THE EIGENVALUES OF LAPLACE AND SCHRÖDINGER OPERATORS ON SUBMANIFOLDS

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1 UNIVERSAL INEQUALITIES FOR THE EIGENVALUES OF LAPLACE AND SCHRÖDINGER OPERATORS ON SUBANIFOLDS AHAD EL SOUFI, EVANS. HARRELL II, AND SAÏD ILIAS Abstract. We establish inequalities for the eigenvalues of Schrödinger operators on compact submanifolds possibly with nonempty boundary) of Euclidean spaces, of spheres, and of real, complex and quaternionic projective spaces, which are related to inequalities for the Laplacian on Euclidean domains due to Payne, Pólya, and Weinberger and to Yang, but which depend in an explicit way on the mean curvature. In later sections, we prove similar results for Schrödinger operators on homogeneous Riemannian spaces and, more generally, on any Riemannian manifold that admits an eigenmap into a sphere, as well as for the Kohn Laplacian on subdomains of the Heisenberg group. Among the consequences of this analysis are an extension of Reilly s inequality, bounding any eigenvalue of the Laplacian in terms of the mean curvature, and spectral criteria for the immersibility of manifolds in homogeneous spaces.. introduction Universal eigenvalue inequalities date from the work of Payne, Pólya, and Weinberger in the 950 s [27], who considered the Dirichlet problem for the Laplacian on a Euclidean domain. In this and similar cases, the term universal applies to expressions involving only the eigenvalues of a class of operators, without reference to the details of any specific operator in the class. Since that time the essentially purely algebraic arguments that lead to universal inequalities have been adapted in various ways for eigenvalues of differential operators on manifolds. E.g., see [2, 7, 8, 5, 6, 2, 22, 26, 30, 33]. For a review of universal eigenvalue inequalities, we refer to [, 3].) In particular, Ashbaugh and Benguria discussed universal inequalities for Laplacians on subdomains of hemispheres in [2], and Cheng and Yang have treated the case of Laplacians on minimal submanifolds of spheres [7, 8]. When either the geometry is more complicated or a potential energy is introduced, analogous inequalities must contain appropriate modifications. Our point of departure is a recent article [5], in which Date: 0 December athematics Subject Classification. 58J50; 58E; 35P5. Key words and phrases. eigenvalues, Laplacian, Schrödinger operator, Reilly inequality, Kohn Laplacian.

2 2 AHAD EL SOUFI, EVANS. HARRELL II, AND SAÏD ILIAS the eigenvalues of Schrödinger operators on hypersurfaces were studied and some trace identities and sharp inequalities were presented, containing the mean curvature explicitly. The goal of the present article is to further study the relation between the spectra of Laplacians or Schrödinger operators and the local differential geometry of submanifolds of arbitrary codimension. The approach is based on an algebraic technique which allows us to unify and extend many results in the literature see [, 3, 4, 5, 6, 2, 22, 26, 27, 32, 33] and Remarks 3., 4. 5.). There is an almost immediate extension of the results of [5] to the case of submanifolds of codimension greater than one, and because of the appearance of the mean curvature, we are able to generalize Reilly s inequality [28, 0,, 2] by bounding each eigenvalue of the Laplacian in terms of the mean curvature. In addition we derive the modifications necessary when the domain is contained in a submanifold of spheres, projective spaces, and certain other types of spaces. Finally, we are able to obtain some universal inequalities in the rather different context of the Kohn Laplacian on subdomains of the Heisenberg group. Let n be a compact Riemannian manifold of dimension n, possibly with nonempty boundary, and let be the Laplace Beltrami operator on. In the case where, Dirichlet boundary conditions apply in the weak sense [9]). For any bounded real valued potential q on, the Schrödinger operator H = + q has compact resolvent see [9, Theorem IV.3.7] and observe that a bounded q is relatively compact with respect to ). The spectrum of H consists of a nondecreasing, unbounded sequence of eigenvalues with finite multiplicities [5, 9]: Spec + q) = {λ < λ 2 λ 3 λ i }. Notice that when = and q = 0, the zero eigenvalue is indexed by, that is λ = 0. To avoid technicalities, we suppose throughout that q is bounded, and that the mean curvature of the submanifolds under consideration is defined everywhere and bounded. Extensions to a wider class of potentials and geometries allowing singularities would not be difficult. 2. Submanifolds of R m In this section is either a closed Riemannian manifold or a bounded domain in a Riemannian manifold that can be immersed as a submanifold of dimension n of R m. The main theorem directly extends a result of [5], in which part I) descends ultimately from a result of H.C. Yang for Euclidean domains [32, 7, 3, 4]: Theorem 2.. Let X : R m be an isometric immersion. We denote by h the mean curvature vector field of X i.e the trace of its second fundamental form). For any bounded potential q on, the

3 UNIVERSAL INEQUALITIES FOR EIGENVALUES 3 spectrum of H = + q with Dirichlet boundary conditions if ) must satisfy, k, I) n λ k+ λ i ) 2 4 λ k+ λ i ) λ i + δ i ) II) + 2 ) λ i ) δ i D nk λ k+ λ i + 2 δ i + D nk, where u i are the L 2 normalized eigenfunctions, δ i := and III) D nk := + 2 ) λ i + 2 ) h 2 q u 2 4 i, ) 2 δ i + 4 ) 4 λ 2 i λ i δ i 0. Theorem 2. can be simplified to eliminate all dependence on u i with elementary estimates such as ) ) h 2 h 2 inf 4 q δ i sup 4 q. 2.) Thus: Corollary 2.. Under the circumstances of Theorem 2., k, I a) n λ k+ λ i ) 2 4 λ k+ λ i ) λ i + δ) II a) λ k+ + 4 ) ) where δ := sup h 2 q. 4 λ i + 4δ n. Corollary 2., proved below, can be restated as a criterion for the immersibility of a manifold in R m : Corollary 2.2. Suppose that {λ i } are the eigenvalues of the Laplace Beltrami operator on an abstract Riemannian manifold of dimension n. If is isometrically immersed in R m, then the mean curvature satisfies for each k. h 2 nλ k+ n + 4) k λ i 2.2)

4 4 AHAD EL SOUFI, EVANS. HARRELL II, AND SAÏD ILIAS Corollary 2.2 is representative of a large family of necessary conditions for immersibility in terms of the eigenvalues of Laplace Beltrami and Schrödinger operators on, which will not be presented in detail in this article. See [5] for various sum rules on which such constraints can be based.) Proof of Theorem 2.. For a smooth function G on, we will denote by G the multiplication operator naturally associated with G. To prove Theorem 2. we first need the following lemma involving the commutator of H and G. Lemma 2.. For any smooth G and any positive integer k one has λ k+ λ i ) 2 [H, G]u i, Gu i L 2 λ k+ λ i ) [H, G]u i 2 L 2.3) 2 This lemma dates from [7, Theorem 5], and in this form appears in [3, Theorem 2. ]. Variants can be found in [5, Corollary 4.3] and [23, Corollary 2.8]. Now, let X,... X m be the components of the immersion X. A straightforward calculation gives [H, X α ]u i = [, X α ] = X α )u i 2 X α u i. It follows by integrating by parts that [H, X α ]u i, X α u i L 2 = Thus [H, X α ]u i, X α u i L 2 = α α On the other hand, we have [H, X α ]u i 2 L = 2 X α 2 u 2 i. X α 2 u 2 i = n u 2 i = n. X α )u i 2 X α u i )) 2. Since X is an isometric immersion, it follows that h = X,, X m ), α X α u i ) 2 = u i 2 and α X α)u i X α u i = h u 2 i = 0. Using all these facts, we get [H, X α ]u i 2 L = h 2 u 2 2 i + 4 u i 2, 2.4) α as in [5]. Then u i 2 = = u i + q)u i ). λ i qu 2 i qu 2 i

5 UNIVERSAL INEQUALITIES FOR EIGENVALUES 5 Using Lemma 2. we obtain ) n λ k+ λ i ) 2 λ k+ λ i ) h 2 4q)u 2 i + 4λ i which proves assertion I) of Theorem 2.. From assertion I) we get a quadratic inequality in the variable λ k+ : kλ 2 k+ λ k ) λ i + 4 δ i )+ + 4 ) λ 2 i + 4 λ i δ i 0 n n n n 2.2) The roots of this quadratic polynomial are the bounds in II). The existence and reality of λ k+ imply statement III). Proof of Corollary 2.. To derive II a) from Theorem 2.II), it is simply necessary to replace δ i by δ, and to note that the quantity D nk is bounded above by + 2 which, since n 2 D nk ) k k λ i ) 2 λ i + λ i + 2δ n ) ) ) k k λ2 i, implies that ) 2 2δ + 8δ n n 2 k λ i = λ 2 i 4δ n 2 n k k λ i, ) 2 λ i + δ, 2n with which the upper bound in II) reduces to the right member of II a). We observe next that Theorem 2. enables us to recover Reilly s inequality for λ 2 of the Laplace Beltrami operator on closed submanifolds [, 28]. Indeed, applying I) with k =, λ = 0 and u = V 2, where V is the volume of, we get λ 2 4 n δ = nv h 2 n h 2. oreover, Theorem 2. allows extensions of Reilly s inequality to higher order eigenvalues. For example, the following corollary can be derived easily from Corollary 2.II a) by induction o. Corollary 2.3. Under the circumstances of Theorem 2., k 2, ) k 4 λ k n + λ + C R n, k) h 2,

6 6 AHAD EL SOUFI, EVANS. HARRELL II, AND SAÏD ILIAS where C R n, k) = 4 and q = 0, 4 n + )k ). In particular, when is closed λ k C R n, k) h ) The explicit value for the generalized Reilly constant C R n, k) given in this corollary is likely far from optimal. We regard the sharp value of C R n, k) as an interesting open problem. In the case of a minimally embedded submanifold of a sphere, Cheng and Yang claim a bound on λ k [7], eq..23)) that scales like k 2 n as in the Weyl law. We conjecture that C R n, k) is sharply bounded by a constant times k 2 n when q = 0 and that when q 0, C R n, k) is correspondingly bounded by a semiclassical expression, as is the case for Schrödinger operators on flat spaces. See, for instance, [3], section 3.5 and [24], part III). In [5] it was argued that simplifications and optimal inequalities are obtained in some circumstances where is a hypersurface and the potential q depends quadratically on curvature, a circumstance that arises naturally in the physics of thin structures [3, 4] and references therein). In this spirit we close the section with some remarks for Schrödinger operators H g := + g h 2, for a real parameter g. As was already observed in [5], in view of 2.), simplifications occur when g =, rendering the quantities δ and δ 4 j given above zero. Corollary 2.4. Assume is closed, h is bounded, and H is of the form H g, where g is an arbitrary real number. The inequalities I), II), and III), in Theorem 2. are saturated i.e., equalities) for all k such that λ k+ λ k, if is a sphere. Proof. We begin with the case of the Laplacian, g = 0, for which the eigenvalues of the standard sphere S n are known [25] to be {ll + n )}, l = 0,,..., with multiplicities for l = 0; n + for l = ; and µ n,l := ) n+l n n+l 2 ) n = nn+)...n+l 2))n+2l ) thereafter. Thus l! λ = 0, λ 2 = = λ n+2 = n, etc., with gaps separating eigenvalues λ k and λ k+ whe = m l=0 µ n,l = n+2m n+m ) n m. For the sphere, δ j = n2, and an exact calculation shows, remarkably, that n λ sphere k+ λ sphere i ) 2 = 4 ) λ sphere k+ λ sphere i ) 4λ sphere i + n 2 : To see this, subtract n k λsphere k+ λ sphere i ) 2 from the expression on the right and multiply the result by n )!. After substitution and simplification, the expression reduces to m l= m l+)n+m+l)2l+n )4ll ) n 2 m l) nm 2 +m ll+3))n+l 2)!, l! which evaluates identically to 0. Algebra was performed with the aid of athematica T.)

7 UNIVERSAL INEQUALITIES FOR EIGENVALUES 7 This establishes equality in I), and consequently II) and III) for this case. If = S n, h 2 = n 2 is a constant, and if gn 2 is added to, then each eigenvalue is shifted by the same amount and the left side of I) is unchanged, as is the first factor in the sum on the right. As for the other factor, it becomes λ i +δ j = ll+n )+gn 2 + n2 4 gn2 and is likewise unchanged. It follows that the case of equality for H g on the standard sphere persists for all g. 3. Submanifolds of spheres and projective spaces Theorem 2., together with the standard embeddings of sphere and projective spaces by means of the first eigenfunctions of their Laplacians, enables us to obtain results for immersed submanifolds of the latter. In what follows, F will denote the field R of real numbers, the field C of complex numbers, or the field Q of quaternions. The m-dimensional projective space over F will be denoted by FP m ; we endow it with its standard Riemannian metric so that the sectional curvature is either constant and equal to F = R) or pinched between and 4 F = C or Q). For convenience, we introduce the integers if F = R df) = dim R F = 2 if F = C 4 if F = Q. and cn) = { n 2, if = S m 2nn + df)), if = FP m. 3.) Theorem 3.. Let be S m or FP m and let X : be an isometric immersion of mean curvature h. For any bounded potential q on, the spectrum of H = g + q with Dirichlet boundary conditions if ) must satisfy, k N, k, I) n λ k+ λ i ) 2 4 λ k+ λ i ) λ i + δ ) i, where where δ i := 4 II) λ k+ + 2 n D nk := h 2 + cn) 4q)u 2 i, ) λ i + 2 δ i + Dnk k + 2 ) + 4 ) λ i + 2 λ 2 i 4 ) 2 δ i λ i δi 0,

8 8 AHAD EL SOUFI, EVANS. HARRELL II, AND SAÏD ILIAS A lower bound is also possible along the lines of Theorem 2.. As in the previous section, the following simplifications follow easily: Corollary 3.. With the notation of Theorem 3. one has, k, λ k+ + 4 ) λ i + 4 n δ, where δ := 4 sup h 2 + cn) 4q). oreover, as in the discussion for Corollary 2.4, when is a submanifold of a sphere or projective space, a simplification occurs in Theorem 3. and Corollary 3. when qx) = 4 h 2 + cn)), in that the curvature and potential do not appear explicitly at all. Remark 3.. Theorems 2. and 3. and corollaries 2. and 3. unify and extend many results in the literature see [, 3, 4, 5, 6, 2, 22, 26, 27, 32, 33] and the references therein). In particular, the recent results of Cheng and Yang [7] and [8] concerning the eigenvalues of the Laplacian on - a domain or a minimal submanifold of S m - a domain or a complex hypersurface of CP m respectively, appear as particular cases of Theorem 3.. Recall that a complex submanifold of CP m is automatically minimal that is, h = 0). Proof of Theorem 3.. We will treat separately the cases = S m and = FP m. Immersed submanifolds of a sphere: Let = S m and denote by i the standard embedding of S m into R m+. We have hi X) 2 = hx) 2 + n 2. Applying Theorem 2. to the isometric immersion i X :, g) R m+, we obtain the result. Immersed submanifolds of a projective space: First, we need to recall some facts about the first standard embeddings of projective spaces into Euclidean spaces see for instance [6, 29, 30] for details). Let m+ F) be the space of m + ) m + ) matrices over F and set H m+ F) = {A m+ F) A := t Ā = A} the subspace of Hermitian matrices. We endow m+ F) with the inner product given by A, B = 2 tra B ). For A, B H m+ F), one simply has A, B = tra B). 2

9 UNIVERSAL INEQUALITIES FOR EIGENVALUES 9 The first standard embedding ϕ : FP m H m+ F) is defined as the one induced via the canonical fibration S m+)d FP m d := df)), from the natural immersion ψ : S m+)d F m+ H m+ F) given by z 0 2 z 0 z z 0 z m ψz) = z z 0 z 2 z z m. z m z 0 z m z z m 2 The embedding ϕ is isometric and the components of ϕ m+ I are eigenfunctions associated with the first eigenvalue of the Laplacian of FP m see, for instance, [30] for details). Hence, ϕfp m ) is a minimal submanifold of the hypersphere S m/2m + )) of H m+ F) centered at m+ I. Lemma 3.. Let X : FP m be an isometric immersion and let h and h be the mean curvature vector fields of the immersions X and ϕ X respectively. Then we have h 2 = h 2 + 4nn + 2) Ke i, e j ) 3 where K is the sectional curvature of FP m and e i ) i n is a local orthonormal frame tangent to X). We refer to [6, 29], or [30] for a proof of this lemma. Now, from the expression of the sectional curvature of FP m, i j we get Ke i, e j ) = if F = R. Ke i, e j ) = + 3 e i Je j ) 2, where J is the almost complex structure of CP m, if F = C. Ke i, e j ) = + 3 r= 3 e i J r e j ) 2, where J, J 2, J 3 ) is the almost quaternionic structure of QP m, if F = Q. Thus in the case of RP m, we obtain h 2 = h 2 + 2nn + ). For CP m, we get i j h 2 = h 2 + 2nn + ) + 2 i,j e i Je j ) 2 = h 2 + 2nn + ) + 2 J T 2 h 2 + 2nn + 2), 3.2) where J T is the tangential part of the almost complex structure J of CP m. Indeed, we clearly have J T 2 n, where the equality holds if and only if X) is a complex submanifold of CP m. For the case of

10 0 AHAD EL SOUFI, EVANS. HARRELL II, AND SAÏD ILIAS QP m, we obtain similarly h 2 = h 2 + 2nn + ) e i J r e j ) 2 i,j r= 3 = h 2 + 2nn + ) + 2 Jr T 2 h 2 + 2nn + 4), 3.3) r= where Jr T ) r 3 are the tangential components of the almost quaternionic structure of QP m. The equality in 3.3) holds if and only if n 0 mod 4) and X) is an invariant submanifold of QP m. To finish the proof of Theorem 3., it suffices to apply Theorem 2. to the isometric immersion ϕ X of in the Euclidean space H m+ F) using the inequalities 3.2) and 3.3). Remark 3.2. It is worth noticing that in some special geometrical situations, the constant cn) in the inequalities of Theorem 3. and corollary 3. can be replaced by a sharper one. For instance, when = CP m and - is odd dimensional, then one can replace cn) by c n) = 2nn + 2 ), n - X) is totally real that is J T = 0), then cn) can be replaced by c n) = 2nn + ). Indeed, under each one of these assumptions, the estimate of J T 2 by n see the inequality 3.2) above) can be improved by elementary calculations. 4. anifolds admitting spherical eigenmaps Let, g) be a compact Riemannian manifold. A map ϕ = ϕ..., ϕ m+ ) :, g) S m is termed an eigenmap if its components ϕ..., ϕ m+ are all eigenfunctions associated with the same eigenvalue λ of the Laplacian of, g). Equivalently, an eigenmap is a harmonic map with constant energy density α ϕ α 2 = λ) from, g) into a sphere. In particular, any minimal and homothetic immersion of, g) into a sphere is an eigenmap. oreover, a compact homogeneous Riemannian manifold without boundary admits eigenmaps for all the positive eigenvalues of its Laplacian see for instance [22]). We still denote by {u i } a complete L 2 -orthonormal basis of eigenfunctions of H associated to {λ i }. Theorem 4.. Let λ be an eigenvalue of the Laplacian of, g) and assume that, g) admits an eigenmap associated with the eigenvalue λ. Then, for any bounded potential q on, the spectrum of H = g + q with Dirichlet boundary conditions if ) must satisfy, k N, k,

11 I) where UNIVERSAL INEQUALITIES FOR EIGENVALUES λ k+ λ i ) 2 II) λ k+ + 2 n ) k ˆD nk = 2n+2) λ k+ λ i ) λ + 4 λ i λ i + λ 4 inf q) 2n + λ i +kλ inf q)) 2 4nkn+4) ˆDnk 2nk. qu 2 i )). λ 2 i +λ inf q)a) Corollary 4.. Let, g) be a compact homogeneous Riemannian manifold without boundary. The inequalities of Theorem 4. hold, λ being here the first positive eigenvalue of the Laplacian of, g). Remark 4.. Theorem 4. and Corollary 4. are to be compared to results of [7, 6, 22]. Proof of Theorem 4.. Let ϕ = ϕ..., ϕ m+ ) :, g) S m be a λ- eigenmap. As in the proof of Theorem 2., we use Lemma 2. with G = ϕ α, α =, 2,..., m +, to obtain λ k λ i ) [H, ϕ α ]u i 2 L. 2 λ k λ i ) 2 [H, ϕ α ]u i, ϕ α u i L 2 α α A direct computation gives [H, ϕ α ]u i = λϕ α u i 2 ϕ α u i and [H, ϕ α ]u i, ϕ α u i L 2 = λ ϕ 2 αu 2 i 2 Summing up, we obtain [H, ϕ α ]u i, ϕ α u i L 2 = λ, α ϕ 2 α u 2 i. since α ϕ2 α is constant. Since α ϕ α 2 = λ and u i 2 = λ i qu2 i, the same kind of calculation yields [H, ϕ α ]u i 2 L = λ ϕ 2 α u i ) 2 α α λ ϕ α 2 u i 2 In conclusion, we have λ λ k+ λ i ) 2 α = λλ + 4λ i qu 2 i ). λ k+ λ i )λ 2 + 4λλ i qu 2 i )),

12 2 AHAD EL SOUFI, EVANS. HARRELL II, AND SAÏD ILIAS which gives the first assertion of Theorem 4.. We derive the second assertion as in the proof of Theorem Applications to the Kohn Laplacian on the Heisenberg group. Let us recall that the 2n + -dimensional Heisenberg group H n is the space R 2n+ equipped with the non-commutative group law x, y, t)x, y, t ) = x + x, y + y, t + t + 2 x, y R n x, y R n), where x, x, y, y R n, t and t R. Its Lie algebra H n has as a basis the vector fields {T = t, X i = + y i x i 2 t, Y i = x i y i 2 t ; i n}. We observe that the only non trivial commutators are [X i, Y j ] = T δ ij, i, j =,, n. Let H n denote the real Kohn Laplacian or the sublaplacian associated with the basis {X,, X n, Y,, Y n }): H n = n X 2 i +Y 2 i = R2n xy + 4 x 2 + y 2 ) 2 t 2 + t n y i x i x i y i We shall be concerned with the following eigenvalue problem : H nu = λu in u = 0 on. 5.) where is a bounded domain of the Heisenberg group H n with smooth boundary. It is known that the Dirichlet problem 5.) has a discrete spectrum. The Kohn Laplacian dates from [20], and the problem 5.) has been studied, e.g., in [8, 26]. We denote its eigenvalues by 0 < λ λ 2 λ k +, and orthonormalize its eigenfunctions u, u 2, S,2 0 ) so that, i, j, u i, u j L 2 = u i u j dx dy dt = δ ij. Here, S,2 ) denotes the Hilbert space of the functions u L 2 ) such that X i u), Y i u) L 2 ), and S,2 0 denotes the closure of C0 ) with respect to the Sobolev norm u 2 S =,2 H nu 2 + u 2 )dx dy dt, with H nu = X u),, X n u), Y u),, Y n u)). We shall prove a result similar to Theorem 2. for the problem 5.): Theorem 5.. For any k ).

13 I) n UNIVERSAL INEQUALITIES FOR EIGENVALUES 3 λ k+ λ i ) 2 2 λ k+ λ i )λ i II) n + nk ) λ i D λ k+ n + nk ) λ i + where D nk = + n) k k λ i D nk ) k n) k λ2 i 0. Remark 5.. Using the Cauchy Schwarz inequality k λ i) 2 k k λ2 i, we deduce from Theorem 5. II) that λ k+ k + 2 ) ) λ i nk which improves a result of Niu and Zhang [26]. Proof. The key observation here is that Lemma 2. remains valid for H = L = H n and G = x α or G = y α. Thus we have n λ k+ λ i ) 2 [L, x α ]u i, x α u i L 2 + [L, y α ]u i, y α u i L 2) α= with Thus, α= n α= n λ k+ λ i ) [L, x α ]u i 2 L + [L, y 2 α ]u i 2 L ) 5.2) 2 [L, x α ] u i = 2X α u i ) and [L, y α ] u i = 2Y α u i ). [L, x α ]u i 2 L + [L, y 2 α ]u i 2 L = 4 2 H nu i 2 = 4λ i. Now, using the skew-symmetry of X α resp. Y α ), we have X α u i ) x α u i = u i X α x α u i ) = u 2 i X α u i )x α u i and the same identity holds with y α and Y α. Therefore, 2 X α u i )x α u i = 2 Y α u i )y α u i = u 2 i =. We put these identities in 5.2) and obtain the first assertion of Theorem 5.. The second assertion follows as in the proof of Theorem 2.. Acknowledgments. This work was partially supported by US NSF grant DS , and was done in large measure while E. H. was a visiting professor at the Université François Rabelais. We also wish to thank ark Ashbaugh and Lotfi Hermi for remarks and references.

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